TopCoder Statistics - Problem Statement

Problem Statement

Teja's college is organizing a tournament in the card game Literature. Teja wanted to take part in the tournament, so he formed a team with Vinay and Sohail. They wanted to practice for the tournament, but no other team wanted to play them - the teams didn't want to leak their strategies before the tournament. But Teja did not give up. He decided that they will practice on their own, and he came up with a game that will help them train their memory.

The practice game is played with 3*N cards. The cards are numbered from 1 to 3*N, inclusive. At the beginning of the game, the cards are shuffled and each player takes N of them.

The game is played in turns. The turns are numbered starting from 0. The players take the turns in the cyclic order "Teja, Vinay, Sohail". That is, Teja takes turn 0, Vinay takes turn 1, Sohail takes turn 2, Teja takes turn 3, and so on.

In each turn, the player selects a random element X from the set of the 2*N cards they don't have, and makes a true statement "I don't have the card number X."

Teja remembers everything that was said during the game. Vinay and Sohail are really bad at the game. They can take valid turns, but they never remember anything the other players said.

You are given the int N. You are also given the int[] Teja with N elements: the numbers on Teja's cards. Finally, you are given the int[] history. There have already been size(history) turns. In turn i the active player announced that he does not have the card number history[i].

Compute and return the expected number of turns from the beginning of the game to the moment when Teja knew the exact distribution of all cards.

Definition

Class:: Literature
Method:: expectation
Parameters:: int, int[], int[]
Returns:: double
Method signature:: double expectation(int n, int[] Teja, int[] history)
(be sure your method is public)

Notes

All random choices are made with uniform probabilities. All random events are mutually independent.
Your return value must have an absolute or a relative error at most 1e-9.

Constraints

N will be between 1 and 1000, inclusive.
Teja will contain exactly N elements.
Each element of Teja will be between 1 and 3*N, inclusive.
All elements of Teja will be distinct.
history will contain between 0 and 200 elements, inclusive.
Each element of history will be between 1 and 3*N, inclusive.
Elements of history will correspond to a valid game.
In particular, for each valid i, history[3*i] will never be one of Teja's cards.

Examples

10

{5,29,12,16,25,17,18,30,27,10}

{4,6,5,23,22,29,20,8,12,3,13,16,1,15,25,4,6,17,23,22,18,20,8,30,3,13,27,1,15,10,4,6,5,23,22,29,20,8,12,3,13,16,1,15,25,4,6,17,22,29,26,8,12,19,13,16,14,15,25,9,6,5,4,22,29,23,8,12,20,13,16,3,15,25,1,6,5,24,22,29,26,8,12,19,13,16,14,15,25,9,6,5,4,22,29,23,8,12,20,13,16,3,15,25,1,6,5,24,22,29,26,8,12,19,13,16,14,15,25,9,6,5,4,22,29,23,8,12,20,13,16,3,15,25,1,6,5,24,22,29,26,8,12,19,13,16,14,15,25,9,6,5,4,22,29,23,8,12,20,13,16,3,15,25,1,6,5}

Returns: 78.0
2

{6,2}

{}

Returns: 11.676190476190474
2

{3,6}

{1}

Returns: 11.676190476190474
1

{3}

{1}

Returns: 3.333333333333333

There has already been one turn. With probability 1/2 there will be exactly one more turn. This happens if the number announced by Vinay in the next turn is not 3, and the probability of that event is exactly 1/2, because Vinay is announcing one of two cards he does not have (chosen uniformly at random). In other scenarios the game will take more than one turn. The final answer is the sum over all possible scenarios of (the probability of that particular scenario) times (the number of turns it took Teja to learn everything in that particular scenario).
1

{3}

{2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2,1,3,2}

Returns: 2.0
10

{1,4,16,22,2,20,10,29,12,14}

{17,8,1,26,28,4,11,27,16,18,6,22,13,5,2,30,23,20,9,21,10,24,25,29,3,7,12,19,15,14,17,8,1,26,28,4,11,27,16,18,6,22,13,5,2,30,23,20,21,10,9,25,29,24,7,12,3,15,14,19,8,1,17,28,4,26,27,16,11,6,22,18,5,2,13,23,20,30,21,10,9,25,29,24,7,12,3,15,14,19,8,1,17,28,4,26,27,16,11,6,22,18,5,2,13,23,20,30,21,10,9,25,29,24,7,12,3,15,14,19,8,1,17,28,4,26,27,16,11,6,22,18,5,2,13,23,20,30,21,10,9,25,29,24,7,12,3,15,14,19,8,1,17,28,4,26,27,16,11,6,22,18,5,2,13,23,20,30,21,10,9,25,29,24,7,12,3,15,14,19,8,1,17,28,4,26,27,16,11,6,22,18,5,2,13,23,20,30,21,10}

Returns: 29.0
1000

{2759,995,2853,1027,191,600,557,416,290,2209,2325,1415,743,2100,1055,2155,1119,934,870,1865,269,1124,1673,1654,194,2060,2142,503,1339,391,119,2145,1435,2852,2274,355,688,2818,2710,1556,1661,1985,1762,2268,759,2595,1587,2605,402,2560,2559,824,1191,2680,2475,1674,2619,2076,64,2123,822,2918,2141,163,2056,2926,1832,1009,452,1801,2072,1297,513,304,983,126,564,2050,1907,1550,291,1271,2616,633,373,1905,1637,1239,1185,1628,2653,1493,2555,2509,1366,774,1636,1172,1300,2151,992,2170,2447,489,2267,2368,1164,2250,505,1619,95,1389,219,422,1750,708,1392,1783,81,1466,2440,2108,2928,1052,2922,1877,295,2679,232,990,2797,484,2887,1204,272,1377,927,2338,654,1714,150,1589,2551,1997,362,1093,92,189,2502,160,2953,101,2113,2791,2573,2958,1992,1402,525,2907,2083,264,844,1849,98,2036,2234,1525,1453,531,2934,1945,2398,1409,1820,1242,1490,1527,1281,1740,594,2402,1265,2303,472,1100,2747,2627,885,1860,1421,994,646,100,1057,2631,1061,162,2580,676,720,2871,2601,2063,27,2205,1386,2096,2266,1024,1045,2392,1831,358,2051,2213,1444,1283,520,211,2001,543,2655,2022,1812,1850,1605,1036,2470,2445,2848,2319,2805,2455,859,2798,811,2228,497,67,595,1900,2344,1883,1643,866,2629,1870,2541,2433,1568,631,748,1692,2557,2237,2134,711,2476,523,623,2378,1163,2404,2825,2385,244,1908,2516,2207,2280,1781,2832,2285,2632,1705,1139,2582,2768,763,2916,2842,2206,113,2381,2152,320,2298,1355,299,770,1759,784,1075,559,1933,2615,973,776,2013,2838,1633,138,1681,985,436,2668,188,1410,1701,1982,2343,1261,260,1351,1299,1043,2521,383,498,1635,2472,2948,2137,2996,137,2542,1647,591,1606,333,2777,169,1785,862,713,2906,1790,625,971,2265,1487,2883,2012,247,2293,1660,1978,347,2843,1866,487,900,404,323,1708,1540,2094,2339,2350,2307,1662,612,685,605,840,421,1340,2957,1780,311,534,1225,2656,2844,2621,2121,2919,1153,571,2429,2452,1226,1775,641,1666,1773,1634,2016,378,779,1372,2675,1202,856,1037,2965,2672,204,2049,1032,215,1307,196,254,1316,2088,2411,389,1352,2340,563,2038,1730,2494,918,2306,1089,2400,49,2822,972,2944,1150,1894,2418,2571,1404,1614,1641,1575,1694,2104,517,2637,2196,2709,2058,1108,120,2183,112,2232,1839,1816,2795,329,1805,1863,608,2413,815,2513,2969,471,2324,1571,1006,1,2181,1274,1401,2741,1983,1019,2497,555,735,1914,1252,1323,1537,2066,481,143,2905,1482,2740,357,473,1732,2927,2093,954,3,1432,573,1070,1236,493,1203,1616,2604,1599,1754,2332,496,1209,2275,5,1233,1506,1349,1196,2020,1304,213,2226,1308,2370,2321,963,1931,1745,1132,1087,466,2708,2539,339,2674,2875,2596,2438,2009,2441,754,130,94,2041,2706,756,537,1176,2177,1786,2295,88,248,445,1980,2175,1999,2271,1756,627,2373,750,1590,2719,2264,2390,2,606,1227,863,576,989,1109,1084,2220,1755,842,2197,2932,738,2533,2282,1020,888,1988,786,1272,2098,2394,1014,1385,1324,1969,156,2570,2662,149,903,1734,1778,151,1951,2290,2903,2023,2337,1213,1677,2689,2239,1821,519,1664,1047,2462,2827,912,804,2657,651,1174,977,32,93,1815,1217,1962,1090,1116,154,2998,542,922,1977,1461,2431,2484,2202,2483,118,2068,2960,1854,1406,1111,1680,1484,1112,1818,317,212,2289,2770,2297,243,781,2600,1184,808,280,237,923,454,1000,805,2391,760,2658,266,1761,1718,2789,397,1038,648,2252,577,31,325,30,1753,2975,1518,1134,790,1819,386,957,566,2454,78,1321,806,2244,1725,1211,1968,915,371,2813,450,872,2576,722,2750,1293,321,1855,1400,401,2742,410,2666,1370,1071,2823,2534,1967,1259,1888,387,1767,136,2090,203,1380,2963,2721,1479,582,65,1041,699,948,2518,1557,1543,728,848,547,1053,2941,2910,2474,698,302,1546,303,2025,1396,1318,783,1126,1160,2701,1147,63,2773,18,1515,2972,1394,1604,201,2568,2597,1106,1489,1686,1294,1223,562,1516,2457,695,1909,1738,1889,258,515,871,981,1114,524,1426,1264,1325,1727,1180,1707,1736,791,236,803,2492,1054,1393,2191,1884,1154,1485,892,2040,2525,1911,1013,1488,400,2861,959,2463,73,2556,478,207,2184,1144,2070,2278,2105,209,2031,1470,190,777,431,821,1441,2210,1902,1360,1562,1553,197,2995,845,2782,1700,2406,2061,867,1437,2763,2661,343,2190,366,704,659,55,1398,1630,1080,342,1670,2065,2479,943,1834,643,2909,2867,2322,474,1423,1544,2977,1507,2649,2780,393,539,1326,2073,2057,1285,873,183,2624,273,2164,1623,717,2942,2945,2161,1609,457,2695,2353,2334,1148,645,1959,238,210,2495,587,752,2442,29,2341,220,1243,1524,2291,2648,2309,2669,2405,1930,2006,2545,619,2737,2085,2659,2466,950,635,919,432,706,2792,139,469,460,2877,1993,1703,1305,2974,2617,176,2242,869,1595,787,657,667,2204,2317,1990,2300,2558,1367,551,819,288,278,1624,2480,2729,251,82,2486,2219,1179,2821,1699,1065,2235,1379,2000,601,1676,2255,1878,560,253,1953,351,2342,736,548,742,798,2819,832,2581,1910,491,1083,2427,1427,813,1724,2725,1892,861,1539,475,2913,1086,2520,1592,2125,1823,1433,2847,2575,1240,1504,2699,1199,1923,166,1958,1764,1356,1659,1551,2671,1655}

{545,1589,1836,2248,1589,721,2720,1000,317,1189,534,2220,2508,534,1626,1867,254,1207,1522,918,1818,403,918,1061,428,918,1346,1791,2465,2187,1791,1610,2197,1376,2257,576,2764,2065,2412,703,2257,608,2241,1676,1479,2248,1805,1119,1295,2065,2378,2720,1000,1729,2248,681,2577,2893,846,2814,2837,911,2187,545,1807,1061,2241,36,2187,2720,1080,1701,703,1487,2145,797,437,1452,858,2463,1848,2110,329,1479,596,233,1195,1491,2465,737,164,2843,454,686,1562,609,1309,536,2034,2478,918,1061,1491,549,1257,1376,1206,1380,403,2130,2127,826,1000,454,1373,1611,567,664,1061,803,1198,2463,2525,2904,521,721,227,534,1444,889,2465,290,891,600,609,167,2192,1236,2471,939,1848,165,2557,2210,2514,2772,2504,545,681,803,889,520,2262,1311,1546,2104,975,653,576,939,2465,1992,2435,798,346,1376,1077,2645,1376,549,1759,1198,2527,1848,403,278,1195,889,2010,19,1768,903,2230,2508,2407,1929,2508,1466,2185,403,900,2710,2651,2020,1202,604,1589,1701,345,1546}

Returns: 40804.93042773051
1000

{2394,1734,671,1324,63,2780,174,808,1585,1608,2667,378,1197,957,2632,876,1323,1018,2371,589,985,2038,2378,54,240,836,2957,2098,669,1245,1880,1492,1114,436,775,2801,1147,1590,1930,933,2493,2139,2660,1391,1252,2807,1869,780,57,1243,2029,582,2480,314,956,1149,1346,1744,2354,1344,762,1546,493,465,2560,536,862,1435,2452,959,1890,1424,516,2711,2291,15,1237,2430,1703,2514,2592,471,1502,185,2100,2675,509,2627,1322,560,1168,1351,1467,1498,2234,2188,774,345,1180,2090,180,2695,2643,1016,1613,232,2744,648,1308,2558,2539,290,194,2153,2080,770,928,1569,1612,1996,2182,445,1352,2941,768,2893,2003,158,2123,936,969,2685,431,889,857,1332,1004,1102,2409,197,2424,724,1133,1062,2282,2825,2628,85,1368,2947,2205,1452,2746,1866,277,1799,220,73,114,1263,874,947,1515,1507,621,1026,1136,2425,1212,1398,251,326,2351,1536,1111,1538,2037,2638,2251,2197,596,1188,152,2619,666,1361,2505,555,2297,313,1858,391,2758,1233,2115,2569,1164,682,1299,2094,2467,1302,157,728,2774,1850,1946,2127,2666,1055,1108,1533,2189,752,1036,1494,2673,2022,1794,2693,742,1196,105,242,1895,355,2404,2258,474,1184,376,698,2633,2664,2723,1526,661,2753,1462,2043,2793,2729,2597,1477,2176,1557,1038,2436,2892,1363,333,473,130,1907,117,322,1640,2054,1362,2461,1131,2840,2731,2150,11,146,1650,998,2520,1718,1033,2134,2177,1396,1117,1788,512,316,1107,2477,447,2903,643,2253,82,1495,1984,784,196,2113,627,700,2218,963,994,2873,1312,1296,1603,2214,2620,1513,188,372,939,599,2316,38,2310,2499,1337,1195,2342,2930,1258,2313,2073,2292,521,2495,1045,1855,911,2626,968,2974,142,1982,145,1389,65,2030,2819,2987,2914,817,490,1404,2135,1580,1305,2750,997,541,367,1736,297,162,1942,1244,579,1219,2603,2052,86,1202,2985,1766,901,2137,2572,2084,1717,1644,922,413,1371,1091,171,791,1826,1331,202,668,2330,1719,608,1894,2362,766,2879,344,2867,414,2028,2120,2498,149,209,2705,343,1970,2858,1836,1239,217,637,208,1439,1916,840,1821,1743,2130,2741,2895,1563,1990,2765,1068,961,2250,654,2125,2016,452,1574,877,833,1374,2864,2682,1853,2759,1737,1372,2921,2220,155,1688,680,1124,2907,2587,1265,734,2117,1480,2598,2614,1778,1684,2531,1768,760,205,324,1106,1094,1347,960,1503,1614,639,1741,930,419,2071,2927,2510,1568,2846,2533,2239,2303,842,203,2884,2345,2623,1379,401,1856,1532,632,1450,2543,119,1578,1632,2542,67,793,2546,488,2398,2518,2103,2248,350,31,1838,1921,1290,295,388,699,912,730,2408,307,2815,2799,1562,1137,868,921,1763,2487,2782,214,2940,906,2266,542,407,80,87,2600,2733,1472,2476,1428,2653,103,126,1545,575,1373,2889,2108,1384,1543,1377,851,2124,1247,141,2869,1732,190,2200,2853,2856,1705,611,2088,2105,1879,1986,2656,1685,2676,2380,704,2897,697,1153,741,1709,1948,292,1328,707,1924,229,231,976,1182,1963,2805,1813,2288,1751,1595,993,1316,2309,807,1534,846,559,1961,721,94,1268,369,571,2837,252,2537,496,946,679,2179,43,2736,1540,282,1097,2490,177,1670,1217,1537,90,470,995,60,1437,1179,1908,1073,1476,506,1941,1815,2645,2721,1325,2439,398,987,2990,2368,1726,2453,594,605,1978,2478,2908,1998,2,658,1954,491,1220,2190,2559,2938,5,1065,871,1155,2158,881,2703,2095,736,1725,1327,3000,1242,24,1236,1659,504,2795,2639,1714,2978,1896,423,1645,702,534,2422,713,1052,1464,1762,2851,2021,1691,1234,614,1620,2324,2444,2905,2637,2195,1919,353,2844,338,2045,714,1750,2065,211,628,2020,1375,2513,2566,765,357,2041,52,2791,955,41,800,2186,2079,1634,254,179,673,1857,2097,920,510,1174,1343,824,143,2132,530,330,183,328,1544,1639,341,2563,796,834,1561,1039,1953,694,1020,19,17,580,2710,892,1646,1041,866,662,2777,2604,709,2944,624,2717,2804,2607,32,1631,756,397,2512,339,2588,1637,2737,2894,1447,1193,1816,2011,649,1560,746,1931,1810,1707,448,1541,505,256,1134,2570,2131,2988,2616,1001,135,1364,656,2972,2111,2747,1122,1787,2270,2460,565,116,2287,2445,226,2650,739,1947,302,2155,2321,331,2128,1712,1177,2922,1012,2692,1488,446,887,166,975,153,823,1819,1594,300,2902,1434,629,1748,1096,951,2762,1112,1571,1080,2192,2168,2243,1889,1876,2806,44,1468,1186,289,160,1030,1833,1021,1285,2278,1935,2686,392,1313,2152,2252,2346,1079,1873,1973,843,903,1092,678,2898,2525,1478,1868,2555,487,1927,2850,364,1669,1839,1456,1050,825,2183,2227,2662,361,2390,261,2451,1146,291,1786,1399,2178,1754,1713,2862,227,482,460,1110,2618,2489,2006,2680,2827,1501,1253,2379,1677,733,1269,2156,144,2391,2261,613,831,1792,1559,1025,2238,359,2708,1139,540,1756,1338,2206,934,1023,1348,2468,1667,1310,938,1385,519,593,352,9,1694,2934,1118,2283,2170,2545,409,2915,585,2999,1160,2249,1782,1158,2066,275,553,2876,253,1681,1676,2387,2612,1411,1228,1173,1887,2557,2792,2491,1256,1964,170,1272,2503,812,551,1267,2416,743,2096,1740,2887,1453,2335,2981,1698,288,1910,2273,535,684,1461,1933,1842,972,2752,1770,298,1104,2517,2918,451,983,2254,1145,2036}

{1224,2668,1710,2486,2278,972,1366,1282,43,745,378,1987,1366,826,372,410,1948,43,2727,2247,972,1607,1578,518,884,1578,1432,603,2278,1448,2336,1690,43,1800,826,495,2336,1773,1710,1027,1773,1214,884,1134,898,173,2874,422,168,1306,43,1366,1023,1663,970,676,1448,1339,1331,1642,2363,2588,2978,2039,2587,1448,1394,1039,2030,2141,1707,1448,2039,1552,2502,1311,1829,784,1181,1773,2054,2024,1778,1710,2363,2170,972,2276,871,90,1394,1391,1748,2466,2988,2113,1556,608,2780,1758,2438,853,136,565,2893,2276,1364,1743,1757,2267,917,2010,2549,850,329,2194,2994,655,1990,881,1181,2278,1201,1523,1890,1108,1116,1686,1603,2363,1039,911,745,1845,1669,147,765,2455,1127,1306,2842,758,1988,2880,2039,2988,230,1877,1153,1748,1224,140,2252,1914,1282,2208,2418,1990,343,2332,1147,2252,125,1608,586,1701,595,419,2960,680,2097,2010,2485,2253,484,506,1193,2426,1559,834,2352,2653,2105,2039,2603,1480,136,2267,1201,136,2067,2978,1701,826,709,2358,1890,1644,845,2492}

Returns: 40820.30908886231
1000

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Returns: 40814.108885317655
1000

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Returns: 40777.03920822476
1000

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Returns: 40786.44587307895
1000

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{505,1674,2970,206,1674,2970,4,653,2766,1437,1650,2970,206,1081,2970,2862,2539,2444,2616,1175,2882,254,1958,2970,2862,1674,2251,2455,2,763,2515,2889,617,857,1329,2217,1393,561,499,679,1674,499,1075,1650,1296,1694,268,189,1633,956,1011,1437,2079,49,1739,2501,1011,743,1660,2342,328,2079,1250,206,2501,2882,254,431,2060,254,1727,2766,2849,409,2158,2849,337,832,1075,1990,2357,787,409,924,616,2249,1342,688,1434,832,1012,2936,907,254,2964,2429,254,2501,2653,2545,1057,1507,128,2539,907,206,1082,1762,128,2121,1250,2112,1674,2882,1012,1175,2704,2070,1223,273,2001,181,854,2112,2745,1891,2008,2282,2919,2862,431,1980,2032,2964,2406,1717,1444,2257,25,892,949,1757,2501,2212,2070,431,1877,2873,2930,1363,1075,2964,1182,1413,339,2970,467,1727,1891,2280,2065,1036,2497,1223,2934,2608,1674,1446,915,2,624,1012,1538,1987,206,142,718,558,2121,2832,206,2964,721,254,1232,2766,2616,2703,2106,2849,2563,2671,254,2100,2614,568,1565,2934,2418,2119}

Returns: 40820.24944753998
1000

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Returns: 40804.76073538205
1000

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Returns: 40820.30908886231
1000

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Returns: 40755.51677363249
1000

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Returns: 40792.58525678431
998

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{2914,1703}

Returns: 40665.34113330508
996

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{961,56}

Returns: 40571.87380923503
982

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{191,440}

Returns: 39921.27975682663
994

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{924,1798}

Returns: 40481.43045254687
1000

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{1426,1722}

Returns: 40761.83237723601
982

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{2256}

Returns: 39919.77982061314
994

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{817}

Returns: 40479.93051556183
986

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{1949}

Returns: 40106.39949888782
997

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{2977}

Returns: 40620.10439978955
999

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{1422}

Returns: 40713.583752574086
980

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{}

Returns: 39826.506611931174
980

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{}

Returns: 39826.506611931174
981

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{}

Returns: 39873.140158553055
999

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{}

Returns: 40713.583752574086
987

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{}

Returns: 40153.06964499064
986

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{2828,2914}

Returns: 40107.89943536059
963

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{969}

Returns: 39034.678077263015
965

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{2725,846,2632}

Returns: 39130.74153856624
968

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{293,583,938}

Returns: 39267.38269351833
974

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{1355,984,1649,542,1428}

Returns: 39548.3341710104
1000

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Returns: 40766.412661514485
1000

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Returns: 40772.56333505206
1000

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Returns: 40763.29099652849
1000

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Returns: 40744.69888647741
1000

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Returns: 40757.11318804945
1000

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Returns: 40753.88657607243
1000

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Returns: 40784.82765347131
1000

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Returns: 40738.46428318747
1000

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Returns: 40778.70727343772
1000

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Returns: 40757.11422390135
1000

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Returns: 40755.51781002015
1000

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{417,997,865,2676,2423,823,1561,1172,428,2925,2030,1408,1171,1590,1137,2054,2609,480,983,1606,4,204,1625,1783,137,2379,1490,264,2101,433,2853,1938,1932,2657,712,2236,579,1936,988,2237,2719,1770,2656,819,688,2336,933,933,557,1310,438,808,2039,1155,430,1386,603,2499,346,872,1865,206,910,2097,1628,774,2637,1342,2470,2413,1981,2818,1273,1118,1422,1388,2160,2454,605,172,2375,1217,2485,1828,2149,1445,1967,757,996,4,1843,244,2915,1213,817,1155,948,2915,2438,877,204,421,2164,1539,1529,851,1527,2054,808,33,1791,2679,1153,837,664,2822,257,2078,2534,2579,1553,2545,2589,1923,1471,957,1200,2793,919,407,2069,2412,2959,1428,55,788,749,2546,734,1529,2093,69,165,1582,1468,651,808,2601,449,2050,2484,2356,1328,146,2705,678,677,2796,2707,1753,445,1263,1372,1561,1276,1652,2757,1129,245,1099,391,2594,1568,889,1782,1815,1361,36,58,2914,2100,530,2477,2737,899,568,2440,831,1504,781,2932,1444,1046,2742,134,1077,1120,1941,921,1908}

Returns: 40736.87606103373
1000

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Returns: 40783.27414302655
1000

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Returns: 40743.01556510539
1000

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Returns: 40777.08729980828
1000

{1129,1324,1415,162,201,700,1498,1286,1163,1952,756,494,1243,1845,2175,309,2934,2511,721,2212,1337,1688,2021,2894,2773,315,662,1733,2632,2116,2203,2472,612,2097,2942,669,34,2725,1903,2156,1545,713,641,2839,2617,1878,2936,1426,2053,2279,606,2532,993,1086,125,1217,2631,1064,749,1928,2276,2237,1328,1227,2133,883,362,1230,2877,298,1123,391,344,1354,2903,235,2326,387,2502,976,1718,448,821,1290,675,2885,310,2477,1624,2404,2266,318,2263,1592,1120,1986,2676,2391,2695,1826,1769,1535,2153,1373,282,2780,306,1071,540,1142,1202,2364,560,2291,715,154,512,1509,1702,1451,1353,613,2383,614,1848,2746,2938,1420,178,1756,817,2941,1470,2124,1824,1526,294,2386,984,2954,2619,2518,2240,1697,142,2462,28,2231,1646,1520,645,2394,658,1399,1908,1821,2409,1479,2262,2215,2086,2027,1979,2056,2190,2248,2708,2064,421,1054,1424,378,1603,297,806,343,130,1322,1443,619,816,1033,2496,1102,1078,1218,2566,2626,159,921,2073,1307,1091,1085,1886,2236,1749,1020,2776,1052,2486,2989,224,581,330,2397,1576,237,2110,1795,1474,2815,66,1068,2466,1753,2858,1511,2296,2868,1493,701,1388,313,740,552,400,2688,2760,406,2028,1488,1804,2480,2865,975,2811,1124,1943,2533,410,1247,336,1946,324,1599,2735,1647,352,83,513,2270,1147,2337,131,1745,364,1216,2863,828,374,2749,2449,2304,32,2172,2892,38,624,736,1510,604,2594,1070,2131,2120,671,1207,1798,1410,2769,2271,2308,1095,2838,481,2497,1018,157,1119,2176,2621,2370,202,1665,292,1491,383,72,2634,2238,1783,2436,58,1969,1741,2506,937,1464,1295,664,1465,2541,1037,1193,636,2835,195,689,2784,1017,1847,111,1185,508,832,1610,190,1343,1113,1447,1914,881,2241,1263,2609,1765,2810,2140,144,472,1806,2900,2614,783,1296,1002,733,1866,879,1446,1436,2654,2412,1522,814,2709,739,898,2742,257,1537,1899,2272,2188,1164,2235,578,2230,1184,957,1241,2939,1897,1976,589,2268,1962,2925,2220,2446,2768,424,456,2824,928,2166,2197,2234,2109,1773,1632,2347,1655,1339,2775,1274,399,1360,327,1910,2878,1092,2831,1542,1223,449,1577,735,1437,1050,215,1250,1907,2888,2853,799,2310,2501,2398,1747,500,1177,2846,2978,1103,2371,22,1485,1128,1626,2437,1547,1677,272,319,2825,1168,2508,2382,1703,1876,2540,2764,2731,1883,232,1467,437,18,12,2734,1041,307,1264,119,785,902,2344,2498,1299,1008,2698,1711,2282,218,2068,651,97,14,62,2821,2948,1840,176,150,2516,35,2713,2896,173,533,2074,896,2293,862,1292,2377,109,1061,198,2601,1270,2755,1057,1673,1304,2799,2042,1588,2907,1309,1432,729,1607,608,692,870,2173,952,2384,478,685,2316,1605,2710,1846,1609,2512,447,1188,2338,1828,1759,185,402,2214,563,1066,1888,1313,1291,2229,2790,1435,1811,1754,2581,707,2544,2208,1383,2038,708,2714,2610,2297,964,1016,230,1790,1779,2422,1314,880,753,1575,1985,985,1505,773,1854,2570,542,1562,2426,394,1729,2917,419,810,2362,2261,2645,960,762,1708,1924,1596,2174,804,263,2093,2969,2071,2855,674,2869,81,546,891,1133,2981,361,597,939,593,1389,2469,2005,1224,2355,179,359,1001,2965,269,670,754,1137,1600,2302,1453,427,1196,1440,1622,1564,2169,1785,2993,501,1386,1919,354,26,770,1363,755,1929,92,291,1524,1036,1512,551,963,845,2683,2232,697,972,1468,857,286,2573,2306,1796,1048,1933,2891,1725,1500,2450,98,1973,2882,1896,305,493,497,1236,2726,2301,1013,377,667,2490,1055,1351,1567,2146,2741,314,795,2322,2186,585,1877,1761,1852,2187,1742,2061,1272,1538,321,789,43,251,788,1189,2435,2305,2239,1289,672,1594,761,1963,1445,666,657,2874,2644,811,2585,1797,1187,2361,1995,2984,1955,1034,2629,1144,2257,1923,2218,1568,299,229,369,2433,2342,316,1856,833,2823,1293,2341,2063,1433,2674,2045,1922,238,1471,1321,1409,1529,2300,67,4,1234,1972,2548,2000,1959,2864,1966,1361,1981,2484,2923,47,1253,1280,1395,2457,2860,2336,2447,300,2121,1104,1195,544,2002,477,502,1090,33,1975,2325,889,417,2791,442,895,2373,1890,2826,1560,2155,1825,2405,16,2164,680,239,471,2213,1698,1394,1573,687,386,1911,584,1557,8,1770,2181,1715,1458,94,1077,850,1606,285,2101,135,1049,977,2194,193,2134,781,2849,1687,763,2630,2380,271,710,407,917,927,1140,1710,505,576,458,2076,2539,326,1311,2505,2915,1850,2255,1987,618,2032,518,1810,2393,1197,365,2694,466,1746,965,1843,2650,752,2638,262,2489,393,180,766,75,632,2320,1553,1763,2030,727,1587,1332,25,1175,1768,2321,2046,1784,2182,1258,2854,175,2036,2354,1760,1156,1228,1074,411,1406,825,1521,2100,694,2258,2696,249,2254,534,1712,884,1162,1108,663,2623,2979,1069,914,455,1841,2443,2737,351,280,2111,2800,2329,80,2365,2072,1466,1502,2307,1893,1087,1518,1838,2432,227,930,1460,2388,1707,2881,240,2444,1913,1364,1833,2759,68,836,2008,1237,2564,2918,273,2678,1330,2368,127,2058,397,2246,638,1898,995,1151,1721,1831,970,1827,462,1398,333,1220,871,1858,2139,120,955,1051,23,7,1282,559,646,2816,967,2616,2771,2202,2245,2787,1869,21,2950,2552,1365,78,798,966,980,2588,469,1739,164,1533,2668,1932}

{1333,1436,1497,1998,528,1807,1616,464,1768,926,2005,696,1416,424,1622,2876,624,747,1136,4,1527,2340,546,751,1550,2969,2158,1671,1285,2385,1579,649,2364,2222,2563,589,1225,1830,2623,894,489,70,1208,2440,1600,567,1328,1687,1411,1976,227,2785,3,1338,2088,500,1162,60,1206,2607,2677,1143,2605,1134,158,713,1618,1841,830,1100,2353,850,2267,2398,551,1405,1505,2310,2703,1286,1699,2286,1773,2541,780,2424,2203,2102,2755,1799,2066,2993,2576,840,692,1513,2406,1991,515,2418,777,522,1817,978,1246,1960,247,1899,2161,735,1896,2060,1908,879,2345,2961,2963,1366,777,321,730,2041,376,779,1090,2404,1145,2292,850,1371,850,1249,775,549,939,312,1741,2520,1621,1232,2580,1683,2002,2148,261,199,2225,2968,813,2701,759,1843,442,2147,2412,2280,2290,2844,2846,2165,1141,2277,2537,237,1692,337,2035,2153,2440,2589,2057,922,104,2899,86,1414,18,598,42,2587,1136,462,1793,526,1529,2456,2867,2151,755,1578,347,2373,244,869,2443,1938,2290,2418,1148,2262}

Returns: 40736.87606103373
2

{1,4}

{3,1,6,6,2,5,2,4,1}

Returns: 6.0

Teja has cards 1 and 4. There have already been nine turn of the game. The first few of them look as follows: Turn 0: Teja announced that he does not have card 3. Turn 1: Vinay announced that he does not have card 1. (Teja already knew this, as this is one of his cards.) Turn 2: Sohail announced that he does not have card 6. (Teja can deduce that Vinay has card 6.) Turn 3: Teja announced that he does not have card 6. Turn 4: Vinay announced that he does not have card 2. Turn 5: Sohail announced that he does not have card 5. After this turn, Teja already knows everything: Vinay's cards are {5, 6} and Sohail's cards are {2, 3}. Thus, the expected number of turns from the beginning of the game to the moment when Teja knew the exact distribution of all cards is always exactly 6. The future rounds do not matter in this case.
10

{5,29,12,16,25,17,18,30,27,10}

{4,6,5,23,22,29,20,8,12,3,13,16,1}

Returns: 133.0880654696573
1000

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000 }

{3000, 2001, 2002 }

Returns: 40757.33319150576
3

{1, 2, 3 }

{4, 4, 5, 4, 4, 5, 4, 4, 5 }

Returns: 26.14090909090909
2

{1, 2 }

{4, 4, 3, 4, 4, 3, 4, 4, 3, 4, 4, 3 }

Returns: 18.285714285714285
5

{1, 2, 3, 4, 5 }

{6, 7, 8, 6, 7, 8 }

Returns: 50.49952347270378

Statistics

Problem Statement for "Literature"

Problem Statement

Definition

Notes

Constraints

Examples